Hostname: page-component-745bb68f8f-f46jp Total loading time: 0 Render date: 2025-02-04T18:14:55.506Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE RELATION BETWEEN THE VEC AND BEKK MULTIVARIATE GARCH MODELS

Published online by Cambridge University Press:  04 April 2008

Robert Stelzer
Robert Stelzer*
Affiliation:
Munich University of Technology
*
Address correspondence to Robert Stelzer, Centre for Mathematical Sciences, Munich University of Technology, Boltzmannstraße 3, D-85747 Garching, Germany; e-mail: rstelzer@ma.tum.de.
Get access Rights & Permissions [Opens in a new window]

Abstract

The question of which multivariate generalized autoregressive conditionally heteroskedastic (GARCH) models in the vec form are representable in the BEKK form is addressed. Using results from linear algebra, it is established that all vec models not representable in the simplest BEKK form contain matrices as parameters that map the vectorized positive semidefinite matrices into a strict subset of themselves. Moreover, a general result from linear algebra is presented implying that in dimension two the models are equivalent, and in dimension three a simple analytically tractable example for a vec model having no BEKK representation is given.

Type
Notes and Problems
Information
Econometric Theory , Volume 24 , Issue 4 , August 2008 , pp. 1131 - 1136
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Bauwens, L., Laurent, S., ’ Rombouts, J.V.K. (2006) Multivariate GARCH models: A survey. Journal of Applied Econometrics 21, 79–109.CrossRefGoogle Scholar
Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307–327.CrossRefGoogle Scholar
Boussama, F. (1998) Ergodicité, mélange et estimation dans les modèles GARCH. Ph.D. Thesis, Université Paris 7, Paris, France.Google Scholar
Boussama, F. (2006) Ergodicité des chaînes de Markov à valeurs dans une variété algébrique: Application aux modèles GARCH multivariés. Comptes Rendus Mathématique. Académie des Sciences. Paris 343, 275–278.CrossRefGoogle Scholar
Choi, M.-D. (1975) Positive semidefinite biquadratic forms. Linear Algebra and Its Applications 12, 95–100.CrossRefGoogle Scholar
Engle, R.F. ’ Kroner, K.F. (1995) Multivariate simultaneous generalized ARCH. Econometric Theory 11, 122–150.CrossRefGoogle Scholar
Li, C.-K. ’ Pierce, S. (2001) Linear preserver problems. American Mathematical Monthly 108, 591–605.CrossRefGoogle Scholar
Li, C.-K., Rodman, L., ’ Semrl, P. (2003) Linear maps on selfadjoint operators preserving invertibility, positive definiteness, numerical range. Canadian Mathematical Bulletin 46, 216–228.CrossRefGoogle Scholar
Loewy, R. (1992) A survey of linear preserver problems—chapter 3: Inertia preservers. Linear and Multilinear Algebra 33, 22–30.CrossRefGoogle Scholar
Pierce, S., Lim, M.H., Loewy, R., Li, C.K., Tsing, N.K., McDonald, B., ’ Beasley, L. (1992) A survey of linear preserver problems. Linear and Multilinear Algebra 33, 1–129.Google Scholar
Ribarits, E.M. (2006) Multivariate modelling of financial time series. Ph.D. Thesis, Institute for Mathematical Methods in Economics, Vienna University of Technology, Vienna, Austria.Google Scholar
Scherrer, W. ’ Ribarits, E. (2007) On the parametrization of multivariate GARCH models. Econometric Theory 23, 464–484.CrossRefGoogle Scholar
Schneider, H. (1965) Positive operators and an inertia theorem. Numerische Mathematik 7, 11–17.CrossRefGoogle Scholar
Størmer, E. (1963) Positive linear maps of operator algebras. Acta Mathematica 110, 233–278.CrossRefGoogle Scholar